STAT 100A HWI Note (1) If all the outcomes are equally likely, then Pr( A ) = | A | / | Ω | . (2) Conditional probability Pr( A | B ) = Pr( A ∩ B ) / Pr( B ). Please show all the necessary steps in your calculations. Please be precise with notation. Problem 1: Suppose we ﬂip a fair coin 4 times independently. (1) What is the sample space? (2) What is the set that corresponds to the event that the number of heads is 2? What is its probability? (3) Let Z i = 1 if the i-th ﬂip is head, and Z i = 0 otherwise, for i = 1 , 2 , 3 , 4. Let X be the number of heads. Express X in terms of Z i . (4) What is the probability distribution of X ? That is, what is Pr( X = k ) for k = 0 , 1 , 2 , 3 , 4? Problem 2: Suppose we roll a fair die twice independently. Let X and Y be the two numbers we get. (1) What is the sample space? Let A be the event that X > 4, and B be the event that Y > 4. What are Pr( A ), Pr( B )? (2) Let C be the event that min( X,Y ) > 4? What is Pr( C )? What is the relationship between Pr( C ) and Pr( A ), Pr( B )? (3) Let D
This is the end of the preview.
access the rest of the document.